The repair price of a contextuality scenario is the minimum number of elementary outcome merges whose simultaneous application restores a global section. On binary scenarios it is an affine reparametrisation of the optimum of Min-Rep / LABEL-COVER_min: pi(e) = MinRep(e) - |Q| for Q the site set. The minimisation is therefore twenty-five years old and is not claimed as new; the contribution is the transfer and what it buys in both directions - an additive hardness of approximation for the repair price, and for the hardness-of-approximation quantity a reading as a physical repair cost that vanishes exactly on the non-contextual scenarios. The identity does not survive past binary arity: at arity three the two quantities can separate (pi = 3 against MinRep_gen - |Q| = 1 on an explicit witness), while equality persists on others including Mermin and GHZ, so strictness is exhibited and not generic. Also given: a monotone integer ledger along scenario growth, an incidence law at the price floor stated as a bounded exhaustion with its converse killed by a shipped witness, a measured separation from contextuality degree (flat where the repair price grades, and undefined on scenarios where the repair price is still defined), a complement to Arkhipov's arrangement theorem in the regime his theorem leaves open (stated as a conjecture), and an exact statement of the domain on which the repair price is identically zero. Two stdlib-only verification engines reproduce every recomputable number on three cross-asserted routes, and the complete pre-release review record - two blind refutation-first referee rounds, the repair log, and the residual confirmation - ships with the paper.
The repair price of a contextuality scenario is the minimum number of elementary outcome merges whose simultaneous application restores a global section. On binary scenarios it is an affine reparametrisation of the optimum of Min-Rep / LABEL-COVER_min: pi(e) = MinRep(e) - |Q| for Q the site set. The minimisation is therefore twenty-five years old and is not claimed as new; the contribution is the transfer and what it buys in both directions - an additive hardness of approximation for the repair price, and for the hardness-of-approximation quantity a reading as a physical repair cost that vanishes exactly on the non-contextual scenarios. The identity does not survive past binary arity: at arity three the two quantities can separate (pi = 3 against MinRep_gen - |Q| = 1 on an explicit witness), while equality persists on others including Mermin and GHZ, so strictness is exhibited and not generic. Also given: a monotone integer ledger along scenario growth, an incidence law at the price floor stated as a bounded exhaustion with its converse killed by a shipped witness, a measured separation from contextuality degree (flat where the repair price grades, and undefined on scenarios where the repair price is still defined), a complement to Arkhipov's arrangement theorem in the regime his theorem leaves open (stated as a conjecture), and an exact statement of the domain on which the repair price is identically zero. Two stdlib-only verification engines reproduce every recomputable number on three cross-asserted routes, and the complete pre-release review record - two blind refutation-first referee rounds, the repair log, and the residual confirmation - ships with the paper.
Contextuality is a central feature distinguishing quantum from classical probability theories, but its operational meaning is often stated only qualitatively. In this Letter, we study a simple information-theoretic question: how much additional contextual information must a classical simulation introduce when it tries to keep a shared internal description fixed across contexts? To make this question precise, we analyze a minimal external-label simulation model in which the remaining context dependence is carried only by an auxiliary label. For this model, we define an obstruction cost as the minimum mutual information between the context and the auxiliary label required to reproduce the observed statistics. We then prove a conservative quantitative lower bound: any linear witness that separates the observed statistics from the zero-obstruction set yields a positive lower bound on this cost. We do not claim that this bound is tight, and we do not claim that the simulation model covers every possible classical architecture. Its role is narrower and more explicit: under fixed shared-state semantics, contextuality can be read as a certificate of irreducible external bookkeeping cost in a simple and well-defined simulation model.
ON THE VERSION NUMBER. "v3" is the version of the NOTE, not of this Zenodo deposit. Versions 1 (February 2026) and 2 were circulated privately and never deposited; v1 contained four errors of substance, which are stated with their repairs in the "What changed" section rather than quietly removed. This is the first Zenodo deposit, so if you see a deposit counter it will not match the document version. The document version is the one that matters for citation. Preprint, version 3 (complete rewrite). This note is the synthesis of the contextuality-obstructions program; the proofs live in the companion papers. The proposal is that the Copenhagen viewpoint, stripped of its rhetoric, is a set of structural statements about one geometric object: the transformation groupoid of measurement contexts (maximal abelian subalgebras) with its canonical U(1) 2-cocycle. In that formulation: local validity, the exact classicality of every single context, is a theorem with an explicit witness; complementarity is the nonvanishing holonomy of the cocycle along closed context loops, which detects exactly the all-versus-nothing sector of contextuality; collapse is pinned by statistics, repeatability and covariance to a one-parameter family per degenerate block whose unique efficient member is the Lueders rule, with a classical twin theorem identifying the same axioms' classical solution as Bayesian conditioning; observer-relativity of facts and the movable cut are theorems of an enlarged observer-context category; and the Frauchiger-Renner argument localizes to an ordinary state-dependent violation, with contextual fraction exactly 1/6, carried by a single CHSH facet, the loop holonomy of the observer switch being provably trivial. ATTRIBUTION. The classification of achievable state-independent values, the even/odd dichotomy and the fixed value d/2, is due to Abramsky, Cercelescu and Constantin (FSCD 2024). It is recast here in certificate language; the priority is theirs and is stated in the text. HARDWARE CAVEAT, STATED UP FRONT. Two signatures were measured on superconducting hardware. The quoted significances, about 56 sigma for the triangle holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented model. They are not systematics-aware, and neither experiment is loophole-free. Read them as consistency checks on real hardware, not as precision measurements. ERRATA. Version 1 of this note contained four errors of substance. Section "What changed" states each of them and its repair. They are recorded rather than quietly removed. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. Each tenet is either a theorem of the companion papers, a machine-verified certificate in the public suite, or is explicitly labeled a conjecture. The paper offers structural characterizations within its stated axioms; it is not an interpretation-free derivation of Copenhagen and it does not claim to resolve the measurement problem. AI-ASSISTANCE DISCLOSURE. This work was carried out with substantial AI assistance. Every load-bearing claim is machine-verified by scripts that ship with the papers, and the author takes full responsibility for the contents. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Paper C: https://doi.org/10.5281/zenodo.21633744 (posted 2026-07-28, after three adversarial review passes; it carries a labelled retraction of an earlier sqrt(2) reading). VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
Preprint. Third paper of the contextuality-obstructions program, alongside Paper A (the canonical class of the MASA context stack), Paper B (the obstruction spectrum of Weyl context families), Paper D (the observer-context category) and a synthesis note. The slogan "contextual holonomy encodes quantum contextuality" is attractive and, as usually stated, false. This paper gives it a sharp form. On the context stack of maximal abelian subalgebras the canonical degree-2 class has order exactly n, and its cocycle holonomy along closed context loops detects EXACTLY the all-versus-nothing sector of contextuality for closed-triple Weyl families at even local dimension. Both containments are strict, each by an explicit certificate: at d=3 the class is nontrivial of order 3 while the achievable state-independent AvN value is 0, and KCBS exhibits state-dependent contextuality where the class shadow vanishes. The Wigner-friend loop generated by H and S splits by level: its projective (Pancharatnam) holonomy is +1 while its determinant-section holonomy is exactly -i, so any interpretational claim about "the" holonomy of that loop must state which one it means. ATTRIBUTION. The classification of state-independent values for qudit Weyl families, the even/odd dichotomy and the fixed value d/2, on which the state-independent side of the equivalence rests, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024). Attainment at general even d is the Raussendorf-Okay-Zurel-Feldmann construction. This paper uses both and rediscovers neither. A RETRACTION IS CARRIED IN THIS PAPER, NOT BURIED. An earlier draft presented the tier-two facets of the d=4 census on a (1/(2*sqrt2))Z grid with a single coefficient magnitude 3/(2*sqrt2), and read the resulting unit-coefficient bounds 4*sqrt2 and 14*sqrt2/3 as placing Tsirelson's sqrt2 inside the tier-two geometry. The whole presentation was a normalization artifact and is withdrawn (Retraction 1): the stored lift has coefficients in {0,+-1} with integer bounds 6 and 7, every facet functional in the census is rational, and Q(sqrt2) is not among the eighteen canonical arithmetic fields. Certificate: d4_arithmetic_profile.py. WHERE THE EVIDENCE IS THIN, STATED IN THE PAPER. Two sweeps quoted here drew no positive instances at all: the d=4 half of the 2,100-family equivalence test, and the 6,500-trial false-certificate sweep, which draws no contextual family at any of d=2,4,8 and therefore establishes the absence of false certificates only vacuously. Both are disclosed where they appear. Theorem 2 is labelled machine-validated rather than proved: the closed form is established end to end for the deletion families treated, while the general closed-triple d=2 statement in compressed coordinates remains open and is recorded as open in the verification ledger. HARDWARE CAVEAT, STATED UP FRONT. The quoted significances, about 56 sigma for the ray-level Wigner-friend holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented witness model and conditional on compatibility and nondisturbance assumptions. They are not systematics-aware and neither measurement is loophole-free; a roughly 3.5 sigma preparation offset on one strand is itself evidence of unmodelled systematics. We also record that the pinned value S = 4.6125 cannot be recomputed from the stored counts (a naive recomputation gives 4.5662, a gap of about 2.7 standard errors); the archived value 4.558 +- 0.018 is the one an independent reader can reproduce from the deposited data. Read these as consistency checks on real hardware, not as precision measurements or certifications. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. It was held back from the first publication batch and released only after three adversarial review passes; what each pass found, and how each item was closed, is recorded in verification/INDEX.md rather than quietly fixed. One defect found there is NOT closed and belongs to the companion Paper B, which cites a false-certificate artifact that is not in fact in the repository; it is recorded rather than left for a reader to discover. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Note: https://doi.org/10.5281/zenodo.21630739 . Code archive: https://doi.org/10.5281/zenodo.21498498 AI-ASSISTANCE DISCLOSURE. Large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance by the scripts named in the text, and the author takes full responsibility for the results, including any errors. VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
Bell's theorem rests on three assumptions: locality, determinism, and measurement independence. Experiments violate Bell inequalities, so at least one assumption must be abandoned. The first two exits are well explored; the third — allowing lawful, partial correlation between measurement settings and hidden variables — is usually dismissed as "superdeterminism" and assumed to require total conspiracy. This paper computes what that third exit actually costs. Hall (2010) priced the bipartite case exactly: 13.81% of measurement independence must be surrendered to reproduce singlet-state correlations. This work extends that program to multipartite GHZ/Mermin scenarios and computes, via exact linear programming, the minimum measurement dependence required to reproduce the full faithful statistics — every correlator and every marginal — for 3 through 13 parties. The floors climb a lawful staircase: 1/3, 2/5, 4/9, 8/17, 16/33, 32/65, all obeying the closed form F = R/(2(R+1)) with R the Mermin violation ratio. A proven universal ceiling shows the required dependence never exceeds 1/2 at any experiment size: quantum statistics never force total superdeterminism. The optimal hidden-variable distributions take a closed physical form — uniform measures on the ground states of the prepared state's frustrated stabilizer Hamiltonian — confirmed out-of-sample on cluster states. The result answers a question Hall posed and left open in 2011, and supplies an answer key against which every published lawful-measurement-dependence program (invariant-set theory, cellular-automaton interpretations, future-input dynamics, all-at-once retrocausality, and the displacement-condensate framework of the present authors) is graded in its own formalism. Operationally, the floors are counterfeiting thresholds for multipartite device-independent certificates. Every exact value is machine-verified on independent hardware and reproducible in under one minute from the linked repository: github.com/RandomInternetPreson/moire-phase-space-sampler (DST_Bell_MI directory). Also available at ai.viXra.org/abs/2607.0049.Version 2 (Oct 2026): closed form proved for all n; cluster-core census with exact certificates; scripts and run logs added.
This is the Technical Supplement — the proof-dense companion and the canonical formal record for the paper's claims. The main paper argues; the supplement proves. Every load-bearing claim traces to a named check function in the APF verification engine, and the supplement is the document a referee should audit first. The supplement is self-contained for Paper 14: every definition, proof, and grade the paper rests on appears in this one document — the substrate primitives and the marginal admissibility floor, proved; the locality and cost-uniqueness lemmas; the Δ calculus with both fences' witnesses written out; the Boolean-defender bridge with its five certified physical hypotheses and the 3-4-5 worked witness computed on the page; the gate pipeline with a gate-by-axiom comparison to the operational-reconstruction literature (Hardy; Chiribella–D'Ariano–Perinotti; Masanes–Müller; Barnum–Wilce; Spekkens; Abramsky–Brandenburger); the native correlation family constructed explicitly, rung by rung; the resolution triad set against standard thermodynamics (Landauer, erasure, environment-assisted recovery); a consolidated readings register listing every claim below [P] with its falsifier and status; a glossary mapping the framework's vocabulary onto standard quantum-information terms; and a worked end-to-end instance — device to interrogation normal form to exact feasibility certificate to priced ledger, with every tolerance explicit. A reproducibility appendix pins every claim to the APF theorem bank at a named GitHub commit and the archived Engine release (DOI 10.5281/zenodo.18529115); the certificates reproduce deterministically in exact rational arithmetic. About the series. Admissibility Physics is a structural realist reconstruction program built on a single commitment: any bounded region of the world can hold only finitely many distinctions. A principle of least enforcement cost then selects, among all the ways those distinctions might be arranged, the configurations a finite world can sustain. From this the content of physics is reconstructed rather than postulated: the Standard Model's gauge group and matter content, the Hilbert-space and Born-rule architecture of quantum mechanics, the geometry and field equations of general relativity, the four laws of thermodynamics, and the large-scale budget of cosmology emerge as different readings of one finite-capacity ledger. The unification is quantitative as well as conceptual: the framework's dimensionless constants appear not as free parameters but as counts read off that ledger — among them the dark-energy fraction ΩΛ = 42/61 and the weak mixing angle sin2θW = 3/13 — fixed, in the strongest cases, to within current measurement. The core derivation runs Papers 0–11; the extension papers carry the individual sectors. Each paper's main text and Technical Supplement is deposited separately; every quantitative claim traces to a named, executable check in the public theorem bank (over 3,800 bank-registered theorems, 48 quantitative predictions); and each result is carried at an explicit, graded level of confidence — from analytically forced to structurally grounded — with the measurements that would refute it named. Admissibility Physics is offered as a candidate, not a finished theory. If the program holds, the laws of physics could not have been otherwise: they are what a finite world permits. Website: admissibilityphysics.com. Engine and theorem bank: https://github.com/Ethan-Brooke/APF-Engine. Full series and latest deposits: Zenodo community admissibility_physics. The Engine — Admissibility Physics: An Executable Constraint Framework and Constants Map — DOI 10.5281/zenodo.18529115. Paper 0 — What Physics Permits: A Constraint-First Framework for Physics — DOI 10.5281/zenodo.18439523 Paper 1 — The Enforceability of Distinction — DOI 10.5281/zenodo.18439200 Paper 2 — Finite Admissibility and the Failure of Global Description — DOI 10.5281/zenodo.18439274 Paper 3 — Entropy, Time, and Accumulated Cost — DOI 10.5281/zenodo.18439363 Paper 4 — Admissibility Constraints and Structural Saturation — DOI 10.5281/zenodo.18439397 Paper 5 — Quantum Structure from Finite Enforceability — DOI 10.5281/zenodo.18439433 Paper 6 — Dynamics and Geometry as Optimal Admissible Reallocation — DOI 10.5281/zenodo.18439445 Paper 7 — A Minimal Quantum of Action from Finite Admissibility — DOI 10.5281/zenodo.18439513 Paper 8 — The Admissibility-Capacity Ledger — DOI 10.5281/zenodo.19721384 Paper 9 — The Geometric Substrate as Cost Structure of Comparison Continuations — DOI 10.5281/zenodo.20041675 Paper 10 — The Calculus of Finite Continuability — DOI 10.5281/zenodo.20041680 Paper 11 — Forced Universality from Capacity-Bounded Admissibility — DOI 10.5281/zenodo.20684198 Paper 12 — The Gauge Sector of Admissibility Physics — DOI 10.5281/zenodo.21212521 Paper 13 — The Minimal Admissibility Core — DOI 10.5281/zenodo.18361446 Paper 14 — The Quantum Story — DOI 10.5281/zenodo.21212188 Paper 16 — Markov Breakdown and the Hard Problems — DOI 10.5281/zenodo.20684207 Paper 18 — The Electroweak Sector as a Capacity Equilibrium — DOI 10.5281/zenodo.20684209 Paper 19 — Large-Angle CMB Suppression as a Finite-Continuability Constraint — DOI 10.5281/zenodo.21199218 Paper 20 — The Enforcement Crystal — DOI 10.5281/zenodo.18531732 Paper 23 — Magnetically Dark Winding-Class Quantum Processors — DOI 10.5281/zenodo.21199220 Paper 24 — The Recruitment-Radius Extension — Foundations — DOI 10.5281/zenodo.20684211 Paper 25 — The Recruitment-Radius Extension: Applications — DOI 10.5281/zenodo.21199222 Paper 28 — Absolute Mass Scales from Electroweak Capacity Saturation — DOI 10.5281/zenodo.20684215 Paper 29 — Plaquette Representation Dominance and Confinement — DOI 10.5281/zenodo.20684218 Paper 30 — A Tube Mechanism for the Lattice Mass Gap — DOI 10.5281/zenodo.20684220 Paper 31 — Osterwalder–Schrader Structure of Lattice Yang–Mills — DOI 10.5281/zenodo.20684222 Paper 32 — Quark Anchors as APF Trace Masses — DOI 10.5281/zenodo.21199226 Paper 33 — Trace-to-Scheme Export Architecture — DOI 10.5281/zenodo.20684224 Paper 35 — The Dark Sector as a Two-Role Capacity Decomposition — DOI 10.5281/zenodo.20684228 Paper 36 — The Missing Floor — DOI 10.5281/zenodo.21199228 Paper 37 — Collapse as Realignment — DOI 10.5281/zenodo.21199231 Paper 38 — Unification as Ledger — DOI 10.5281/zenodo.21199233 Paper 39 — Coherent Phase Regimes in the APF Interface Engine — DOI 10.5281/zenodo.21199235 Paper 40 — The Thermodynamics of Finite Distinction (Between Symmetry and the Void) — DOI 10.5281/zenodo.20684235 Paper 41 — The Horizon as a Continuation Ledger — DOI 10.5281/zenodo.20684241 Paper 42 — The Weak Mixing Angle Is Not Free — DOI 10.5281/zenodo.20684245 Paper 43 — The Neutrino Sector — DOI 10.5281/zenodo.21194191 Paper 44 — What a Finite Theory Owes the Continuum — DOI 10.5281/zenodo.21195874 Paper 45 — What the Vacuum Can Keep — DOI 10.5281/zenodo.21194195 Paper 46 — The Strong CP Angle Is Not Free — DOI 10.5281/zenodo.21194197 Paper 47 — The Confinement Antecedent Is Not Forced — DOI 10.5281/zenodo.21194199 Paper 48 — The Light-Bending Weight Is Not Free — DOI 10.5281/zenodo.21199627 Technical Supplements cross-link to their main paper via isSupplementTo and to the engine via isDocumentedBy. 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